Partial Differential Equations

• Introduction– Deng Li

• Discretization Methods– Chunfang Chen, Danny Thorne, Adam

Zornes

CS521 Feb.,7, 2006

What do You Stand For?

• A PDE is a Partial Differential Equation

• This is an equation with derivatives of at least two variables in it.

• In general, partial differential equations are much more difficult to solve analytically than are ordinary differential equations

What Does a PDE Look Like

• Let u be a function of x and y. There are several ways to write a PDE, e.g.,

– ux + uy = 0

– u/x + u/y = 0

The Baskin Robin’s esq Characterization of PDE’s

• The order is determined by the maximum number of derivatives of any term.

• Linear/Nonlinear– A nonlinear PDE has the solution times a

partial derivative or a partial derivative raised to some power in it

• Elliptic/Parabolic/Hyperbolic

Six One Way

• Say we have the following: Auxx + Buxy + Cuyy + Dux + Euy + F = 0.

• Look at B2 - 4AC– < 0 elliptic – = 0 parabolic – > 0 hyperbolic

Or Half a Dozen Another

• A general linear PDE of order 2:

• Assume symmetry in coefficients so that A = [aij] is symmetric. Eig(A) are real. Let P and Z denote the number of positive and zero eigenvalues of A. – Elliptic: Z = 0 and P = n or Z = 0 and P = 0..

– Parabolic: Z > 0 (det(A) = 0).

– Hyperbolic: Z=0 and P = 1 or Z = 0 and P = n-1.

– Ultra hyperbolic: Z = 0 and 1 < P < n-1.

aijuxi x j biuxi cu di1

n

i, j1

n

.

Elliptic, Not Just For Exercise Anymore

• Elliptic partial differential equations have applications in almost all areas of mathematics, from harmonic analysis to geometry to Lie theory, as well as numerous applications in physics.

• The basic example of an elliptic partial differential equation is Laplace’s Equation

– uxx - uyy = 0

The Others

• The heat equation is the basic Hyperbolic – ut - uxx - uyy = 0

• The wave equations are the basic Parabolic– ut - ux - uy = 0

– utt - uxx - uyy = 0

• Theoretically, all problems can be mapped to one of these

What Happens Where You Can’t Tell What Will Happen

• Types of boundary conditions– Dirichlet: specify the value of the function on a

surface

– Neumann: specify the normal derivative of the function on a surface

– Robin: a linear combination of both

• Initial Conditions

Is It Worth the Effort?

• Basically, is it well-posed?– A solution to the problem exists.

– The solution is unique.

– The solution depends continuously on the problem data.

• In practice, this usually involves correctly specifying the boundary conditions

Why Should You Stay Awake for the Remainder of the Talk?

• Enormous application to computational science, reaching into almost every nook and cranny of the field including, but not limited to: physics, chemistry, etc.

Example

• Laplace’s equation involves a steady state in systems of electric or magnetic fields in a vacuum or the steady flow of incompressible non-viscous fluids

• Poisson’s equation is a variation of Laplace when an outside force is applied to the system

Poisson Equation in 2D

Example: CFD